Ramanujan's function k

DefinitionRamanujan's function \(k(q)\)#✓fully proved

For a complex number \(q\), Ramanujan's function is

\[ k(q) = \frac{q\,(q;q^{10})_\infty (q^2;q^{10})_\infty (q^8;q^{10})_\infty (q^9;q^{10})_\infty} {(q^3;q^{10})_\infty (q^4;q^{10})_\infty (q^6;q^{10})_\infty (q^7;q^{10})_\infty}, \]

where \((a;x)_\infty\) is the raw infinite q-Pochhammer product \(\prod_{n \ge 0} (1 - a x^n)\) (equal to \(1\) by convention when this product does not converge unconditionally), and a quotient by \(0\) is read as \(0\).

For \(|q| < 1\) this is the product representation of \(k(q) = r(q)\,r(q^2)^2\), where \(r\) is the Rogers–Ramanujan continued fraction.

noncomputable def ramanujanK (q : ℂ) : ℂ :=
  q * (qPochhammerInfRaw q (q ^ 10) * qPochhammerInfRaw (q ^ 2) (q ^ 10) *
        qPochhammerInfRaw (q ^ 8) (q ^ 10) * qPochhammerInfRaw (q ^ 9) (q ^ 10)) /
    (qPochhammerInfRaw (q ^ 3) (q ^ 10) * qPochhammerInfRaw (q ^ 4) (q ^ 10) *
        qPochhammerInfRaw (q ^ 6) (q ^ 10) * qPochhammerInfRaw (q ^ 7) (q ^ 10))

Sources

• Russelle Guadalupe, New Proofs of Certain Theta Function Identities via Ramanujan's Function, INTEGERS 26 (2026), Article A41, Section 1 (product representation of k(q)).

Uses 1 · Used by 4

Statement usesRaw infinite q-Pochhammer product

Used byRamanujan's function \(k\) at \(q\) and \(q^2\) as an eta quotient · Eta-quotient expressions for Ramanujan's function \(k\) · Modular equation of degree 2 for Ramanujan's function \(k\) · Ramanujan's function \(k\) vanishes at \(0\)

TheoremRamanujan's function \(k\) vanishes at \(0\)#✓fully proved

Ramanujan's function satisfies \(k(0) = 0\).

theorem ramanujanK_zero : ramanujanK 0 = 0

Sources

• Russelle Guadalupe, New Proofs of Certain Theta Function Identities via Ramanujan's Function, INTEGERS 26 (2026), Article A41, Section 1 (product representation of k(q)).

Uses 2

Statement usesRamanujan's function \(k(q)\)

Proof usesRaw infinite q-Pochhammer product

TheoremEta-quotient expressions for Ramanujan's function \(k\)#●ready to prove

Let \(q\) be a complex number with \(0 < |q| < 1\), write \(f_m = (q^m;q^m)_\infty = \prod_{n \ge 1} (1 - q^{mn})\) for \(m \ge 1\), and let \(k = k(q)\) be the value of Ramanujan's function. Then

\[ \frac{f_2 f_5^5}{q f_1 f_{10}^5} = \frac{1}{k} - k, \qquad \frac{f_2^4 f_5^2}{q f_1^2 f_{10}^4} = \frac{1}{k} + 1 - k, \qquad \frac{f_1^3 f_5}{q f_2 f_{10}^3} = \frac{1}{k} - 4 - k. \]

public theorem_wanted ramanujanK_eta_quotients (q : ℂ) (hq : ‖q‖ < 1) (hq0 : q ≠ 0) :
    qPochhammerInfRaw (q ^ 2) (q ^ 2) * qPochhammerInfRaw (q ^ 5) (q ^ 5) ^ 5 /
        (q * qPochhammerInfRaw q q * qPochhammerInfRaw (q ^ 10) (q ^ 10) ^ 5) =
      1 / ramanujanK q - ramanujanK q ∧
    qPochhammerInfRaw (q ^ 2) (q ^ 2) ^ 4 * qPochhammerInfRaw (q ^ 5) (q ^ 5) ^ 2 /
        (q * qPochhammerInfRaw q q ^ 2 * qPochhammerInfRaw (q ^ 10) (q ^ 10) ^ 4) =
      1 / ramanujanK q + 1 - ramanujanK q ∧
    qPochhammerInfRaw q q ^ 3 * qPochhammerInfRaw (q ^ 5) (q ^ 5) /
        (q * qPochhammerInfRaw (q ^ 2) (q ^ 2) * qPochhammerInfRaw (q ^ 10) (q ^ 10) ^ 3) =
      1 / ramanujanK q - 4 - ramanujanK q

Sources

• Russelle Guadalupe, New Proofs of Certain Theta Function Identities via Ramanujan's Function, INTEGERS 26 (2026), Article A41, Lemma 2, Equations (5)-(7).

Uses 2

Statement usesRaw infinite q-Pochhammer product · Ramanujan's function \(k(q)\)

TheoremRamanujan's function \(k\) at \(q\) and \(q^2\) as an eta quotient#●ready to prove

For every complex number \(q\) with \(0 < |q| < 1\), Ramanujan's function \(k\) satisfies

\[ \frac{k(q)}{k(q^2)} - \frac{k(q^2)}{k(q)} = \frac{(q;q)_\infty (q^5;q^5)_\infty^3}{q\,(q^{10};q^{10})_\infty^4}, \]

where \((a;x)_\infty = \prod_{n \ge 0} (1 - a x^n)\) is the infinite q-Pochhammer product.

public theorem_wanted ramanujanK_div_sub_div_sq (q : ℂ) (hq : ‖q‖ < 1) (hq0 : q ≠ 0) :
    ramanujanK q / ramanujanK (q ^ 2) - ramanujanK (q ^ 2) / ramanujanK q =
      qPochhammerInfRaw q q * qPochhammerInfRaw (q ^ 5) (q ^ 5) ^ 3 /
        (q * qPochhammerInfRaw (q ^ 10) (q ^ 10) ^ 4)

Sources

• Russelle Guadalupe, New Proofs of Certain Theta Function Identities via Ramanujan's Function, INTEGERS 26 (2026), Article A41, Lemma 3.

Uses 2

Statement usesRaw infinite q-Pochhammer product · Ramanujan's function \(k(q)\)

TheoremModular equation of degree 2 for Ramanujan's function \(k\)#●ready to prove

For every complex number \(q\) with \(|q| < 1\), the values \(X = k(q)\) and \(Y = k(q^2)\) of Ramanujan's function satisfy \(X^2 - Y + 2XY + X^2 Y + Y^2 = 0\).

public theorem_wanted ramanujanK_sq_relation (q : ℂ) (hq : ‖q‖ < 1) :
    ramanujanK q ^ 2 - ramanujanK (q ^ 2) + 2 * ramanujanK q * ramanujanK (q ^ 2) +
        ramanujanK q ^ 2 * ramanujanK (q ^ 2) + ramanujanK (q ^ 2) ^ 2 = 0

Sources

• Russelle Guadalupe, New Proofs of Certain Theta Function Identities via Ramanujan's Function, INTEGERS 26 (2026), Article A41, Theorem 4.

Uses 1

Statement usesRamanujan's function \(k(q)\)

TheoremRamanujan's level-10 theta function identities#●ready to prove

For complex \(x\) with \(|x| < 1\), let \(\varphi(x) = \sum_{n \in \mathbb{Z}} x^{n^2}\), \(\psi(x) = \sum_{n \ge 0} x^{n(n+1)/2}\) and \(f(-x) = \prod_{n \ge 1} (1 - x^n)\), so that \(f(x) = \prod_{n \ge 1} (1 - (-x)^n)\). Then for every complex number \(q\) with \(|q| < 1\),

\[ \frac{\varphi(q)^5}{\varphi(q^5)} + \frac{\varphi(-q^2)^5}{\varphi(-q^{10})} + 2\,\frac{f(q)^5}{f(q^5)} = 4\,\frac{\psi(q)^5}{\psi(q^5)} \]

and

\[ \varphi(q)^2 \varphi(q^5)^2 - \varphi(-q)^2 \varphi(-q^5)^2 - 16 q^3 \psi(q^2)^2 \psi(q^{10})^2 = 8 q\, f(-q^2)^2 f(-q^{10})^2. \]

public theorem_wanted ramanujan_theta_level_ten (q : ℂ) (hq : ‖q‖ < 1) :
    ramanujanThetaRaw q q ^ 5 / ramanujanThetaRaw (q ^ 5) (q ^ 5) +
        ramanujanThetaRaw (-q ^ 2) (-q ^ 2) ^ 5 / ramanujanThetaRaw (-q ^ 10) (-q ^ 10) +
        2 * (qPochhammerInfRaw (-q) (-q) ^ 5 / qPochhammerInfRaw (-q ^ 5) (-q ^ 5)) =
      4 * (ramanujanThetaRaw q (q ^ 3) ^ 5 / ramanujanThetaRaw (q ^ 5) ((q ^ 5) ^ 3)) ∧
    ramanujanThetaRaw q q ^ 2 * ramanujanThetaRaw (q ^ 5) (q ^ 5) ^ 2 -
        ramanujanThetaRaw (-q) (-q) ^ 2 * ramanujanThetaRaw (-q ^ 5) (-q ^ 5) ^ 2 -
        16 * q ^ 3 * ramanujanThetaRaw (q ^ 2) ((q ^ 2) ^ 3) ^ 2 *
          ramanujanThetaRaw (q ^ 10) ((q ^ 10) ^ 3) ^ 2 =
      8 * q * qPochhammerInfRaw (q ^ 2) (q ^ 2) ^ 2 * qPochhammerInfRaw (q ^ 10) (q ^ 10) ^ 2

Sources

• Russelle Guadalupe, New Proofs of Certain Theta Function Identities via Ramanujan's Function, INTEGERS 26 (2026), Article A41, Theorem 1, Equations (3) and (4).

Uses 2

Statement usesRaw infinite q-Pochhammer product · Raw Ramanujan theta sum

2 fully proved4 ready to prove

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